Shadowing and the continuity of omega-limit sets

This paper examines the relationship between shadowing phenomena and the continuity properties of $\omega$-limit sets in dynamical systems. We give a necessary and sufficient condition for a shadowable point to be an upper (resp. a lower) semicontinuity point of $\omega$-limit sets. Assuming global shadowing, we show that the lower semicontinuity of $\omega$-limit sets is equivalent to the chain continuity. We also show that the lower semicontinuity of $\omega$-limit sets is equivalent to the chain continuity in a general setting. Several examples are given to illustrate the results.